When Small Numbers Lead to Big Errors

As the U.S. military embarks on its review of Don’t Ask, Don’t Tell, to be delivered in a final report later this year, the question arises: How many service members are affected by the policy? To help answer that question, the Pentagon this past summer surveyed its troops, asking them if they served or had ever served with someone they believe to be gay. Leaving aside an obvious problem with the survey—that it is based on pure speculation—it also raises a common statistical challenge: asymmetry in population sizes. Because the vast majority of service members are heterosexual, many more straights will be misclassified as gays than vice versa.

This is a general problem in survey research. For example, Harvard University researcher David Hem­en­way has shown how some well-publicized studies have overestimated the number of guns used in the U.S. for self-defense by 10 times. Even if only 1 percent of respondents answer the survey incorrectly, the error is large compared with the proportion of the general population in any given year that uses guns in self-defense, which reasonable studies show to be about 0.1 percent.* In other words, the misclassification rate far exceeds the actual population size. To get around this problem, we would be wiser to trust surveys of crime victims, which restrict the gun use question to a smaller pool of subjects.

As to our original question, a reasonable (though still imperfect) way to figure out what percentage of military service members are gay is by combining two estimates: the percentage of gays in the general population (easy enough to estimate from national surveys) and an estimate of the percentage of individuals in same-sex unmarried partner couples who report ever having served in the military (known as a probability). By extrapolating from the general population to service members, you are restricting your analysis to same-sex unmarried couples and thus narrowing the pool of potential false positives. Gary J. Gates of the University of California, Los Angeles, estimates using this method that 1.5 percent of men and 6.2 percent of women in the military are gay or bisexual.

*Correction (10/19/10): This sentence was edited after posting. It originally stated that the 1 percent error is large compared with the proportion of the general population that owns guns for self-defense.

ABOUT THE AUTHOR(S)

Gelman is a professor of statistics and political science at Columbia University.